Page 25 - Determinants and Their Applications in Mathematical Physics

10 2. A Summary of Basic Determinant Theory

m m m

= ··· C 1k 1 ··· C jk j ··· C nk n n .

k 1 =1 k 2 =1 k n =1
The function on the right is the sum of m determinants. This identity
n
can be expressed in the form

m m

(k j )
a = a .
(k)

ij ij
n

k=1 k 1 ,k 2 ,...,k n =1
n
h. Column Operations. The value of a determinant is unaltered by adding
to any one column a linear combination of all the other columns. Thus,
if
n

C = C j + k j =0,
j k r C r
r=1
n

= k r C r , k j =1,
r=1
then

C 1 C 2 ··· C ··· C n = C 1 C 2 ··· C j ··· C n .

j
C should be regarded as a new column j and will not be confused

j
with the derivative of C j . The process of replacing C j by C is called a

j
column operation and is extensively applied to transform and evaluate
determinants. Row and column operations are of particular importance
in reducing the order of a determinant.
Exercise. If the determinant A n = |a ij | n is rotated through 90 in the
◦
clockwise direction so that a 11 is displaced to the position (1,n), a 1n is dis-
placed to the position (n, n), etc., and the resulting determinant is denoted
by B n = |b ij | n , prove that
b ij = a j,n−i
B n =(−1) n(n−1)/2 A n .
2.3.2 Matrix-Type Products Related to Row and Column
Operations

The row operations
3

R = u ij R j , u ii =1, 1 ≤ i ≤ 3; u ij =0, i>j, (2.3.1)

i
j=i